NBS Reference Publi- cations NATL INST. OF STAND & TECH AlllDb 3MD173 iiM- NBS TECHNICAL NOTE 1 047 U.S. DEPARTMENT OF COMMERCE / Notionol Bureau of Standards Deconvolutlon of Time Domain Waveforms in the Presence of Noise -QC- 1 100 U5753 NATIONAL BUREAU OF STANDARDS The National Bureau of Standards' was established by an act ot Congress on March 3, 1901. The Bureau's overall goal is to strengthen and advance the Nation's science and technology and facilitate their effective application for public benefit. To this end, the Bureau conducts research and provides: (1) a basis for the Nation's physical measurement system, (2) scientific and technological services for industry and government, (3) a technical basis for equity in trade, and (4) technical services to promote public safety. The Bureau's technical work is per- formed by the National Measurement Laboratory, the National Engineering Laboratory, and the Institute for Computer Sciences and Technology. 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'Headquarters and Laboratories at Gaithersburg, MD, unless otherwise noted; mailing address Washington, DC 20234. -Some divisions within the center are located at Boulder, CO 80303. noMAL feasxAV or JiTAMDAXDS UBKABT OtC 1 1981 "^ Deconvolution of ;;jr Time Domain Woveforms "™ in ihe Presence of Noise ^^ N. S. Nahman M. E. Guillaume Electromagnetic Technology Division National Engineering Laboratory National Bureau of Standards Boulder, Colorado 80303 M£. U.S. DEPARTMENT OF COAAMERCE, Malcolm Doldrige, Secrerory NATIONAL BUREAU OF STANDARDS, Ernesr Ambler, Direcror Issued October 1981 NATIONAL DUP>EAU OF STANDARDS TECHNIGM NOTE 1 047 Nor. Dur, Srond. (U.5.), Tech. Nore 1047, 122 pages (Ocrober 1981) CODEN: NDTNAE US GOVERNMENT PRINTING OFFICE WASHINGTON: 1981 CONTENTS Page FOREWORD V 1. INTRODUCTION - 1 2. GENERAL PRESENTATION OF THE DECONVOLUTION PROBLEM 4 2.1 An Exact Solution Exists for Exact Waveforms 4 2.2 Experimental Situation Yields Inexact Waveforms 4 2.3 The Regularization Operator 5 2.4 Possible Methods of Deconvolution 6 2.4.1 Model Fitting 6 2.4.2 Time Domain Processing 6 2.4.3 Transform Domain Processing , 6 2.5 Loss of Information in the Convolution Process 6 3. EQUIPMENT AND METHODS USED IN THE EXPERIMENTS H 3.1 Acquisition of Waveforms H 3.2 Interactive Processing 11 3.3 Qualitative Measurement of the Quality of One Solution 11 3.4 Quantitative Criteria 11 4. FREQUENCY (TRANSFORM) DOMAIN DECONVOLUTION 15 4.1 Introduction 4.2 Methods Considered 17 4.3 The Two Parameter Method 18 4.3.1 The Filter Structure 18 4.4 Application of the Two Parameter Method 19 4.4.1 Procedural Summary 19 4.4.2 The Estimation of the Insertion Impulse Response of 300 Meters of RG 58 C/U Coaxial Cable 19 4.4.3 The Estimation of the Insertion Impulse Response of a Broadband Antenna ... 21 4.5 On the Nature of the Error, e(k) 22 4.5.1 Error in the Absence of the Filtering Operation and Computation Noise .... 22 4.5.2 Error in the Presence of the Filtering Operation and in the Absence of Computation Errors 22 4.5.3 Error in the Presence of the Filtering Operation and Computation Errors ... 25 4.6 The One Parameter Method 28 4.6.1 The Filter Structure 28 4.6.2 Conversion of a Step-Like Waveform into a Duration Limited One 35 4.6.3 Application of the One Parameter Method 37 4.6.3.1 Procedural Summary 37 4.6.3.2 Estimation of the Insertion Impulse Response of 300 Meters of RG 58 C/U Coaxial Cable 38 4.6.3.3 The Insertion Impulse Response of a Filter for a Step-like, 100 ps Transition Duration, Reference Waveform Generator 39 Page 5. TIME DOMAIN DECONVOLUTION 78 5.1 Introduction 78 5.2 Discrete Classical Time Domain Deconvolution 78 5.2.1 Convolution 78 5.2.2 Z-transform Representation 79 5.2.3 Deconvolution 80 5.2.4 The Effect of Errors in Classical Deconvolution 83 5.2.5 The Critical Dependence of Classical Deconvolution on the First Point in x(k). 85 5.3 Discrete Iterative Time Domain Deconvolution 87 5.3.1 Iterative Method 87 5.3.2 The Effect of Errors in Iterative Deconvolution 89 5.4 Time Domain Regularization 90 5.4.1 Averaging Filter 90 5.4.2 Composite Filtering 91 5.4.2.1 Introduction 91 5.4.2.2 Filter Description 92 5.4.2.3 Frequency Domain Analysis of the Composite Filter 93 5.4.3 Application of Time Domain Filtering 94 5.4.3.1 Practical Procedure 94 5.4.3.2 Estimation of the Insertion Impulse Response of 300 Meters of RG 58 C/U Coaxial Cable 95 6. SUMMARY AND CONCLUSIONS 112 5.1 Summary 112 6.2 Conclusions 112 7. REFERENCES 114 FOREWORD The work reported here is the result of a continuing collaboration between the Centre National D'Etudes Des Telecommunications, France, and the National Bureau of Standards. From October 1978 through September 1979, Dr. Michele Guillaume was a Guest Scientist in the Time Domain Metrology Group of the Electromagnetic Technology Division, NBS , Boulder Laboratories, Boulder, Colorado. At that time Drs. Guillaume and Nahman worked together to produce the experimental results reported here. It has taken some time for the authors to complete their analyses of the experimental data, to provide some analytic bases for understanding the deconvolution and filtering processes, and to prepare this manuscript. All of these latter activities were accomplished with the authors located in France and the USA, respectively. Thus, there has been some delay in publishing this work. DECONVOLUTION OF TIME DOMAIN WAVEFORMS IN THE PRESENCE OF NOISE N. S. Nahman and M. E. Guillaume t Electromagnetic Technology Division National Bureau of Standards Boulder, Colorado 80303 Deconvolution or inverse filtering was used to determine the impulse response of a system using noisy input and output time domain sequences (discrete data) Frequency and time domain methods were studied along with the synthesis of the filters required to obtain stable and smooth results. For the methods studied it was concluded that the superior technique was provided by an optimal frequency domain method implemented via the FFT. Also, it is pointed out that the time domain methods are only in their infancy and still retain the promise of avoiding transform domain filtering. Examples are presented in which the impulse responses are determined in the presence of varying degrees of noise for a coaxial transmission line, a wave-shaping filter, and a broad-band antenna. Key words: deconvolution; impulse response^ inverse filtering; noise contamination; time domain; waveforms. 1. INTRODUCTION well known. If K(t) The analogy between a physical linear system and the convolution equation is h(t), the output waveform is described is the input waveform to a linear device whose impulse response is using the convolution equation, (1-1) ^(t) = fx(T) h(t-T)dr =J x(t-T)h(T) dx or in simplified notation 7(t) = x(t) * h(t) <;i-2) where "*" denotes convolution. The Laplace transform of y(t) is given by Y(s) = X(s) H(s) (^-3) s-domain. The decon- which shows that the convolution process transforms to a product operation in the when either x(t) volution process is defined as the solution of the convolution integral equation (1-1) or h(t) is unknown, i.e.. (1-^) h(t) = y(t) (1/*) x(t) (1-5) H(s) = Y(s) ^ X(s) t Centre de Microelectronique de Grenoble, Centre National d' Etudes de Telecommunications, Meylan, France.